Prove that there exists only one pair $ (p,q) $ of odd primes satisfying the properties that $ p^2\equiv 4\pmod q $ and $ q^2\equiv 1\pmod p. $ [i]Ana Maria Acu[/i]

Problem 7. On the board, Sabir writes 10 consecutive numbers. For each number, Salim writes the sum of its digits on his paper, and Sabrina writes the number of its divisors on her paper. Is it possible for Sabrina’s 10 numbers to be exactly the same as Salim’s 10 numbers in some order? (the repetitions of the numbers should also be the same)

An infinite sheet of paper is divided into equal squares, some of which are colored red. In each $2\times3$ rectangle, there are exactly two red squares. Now consider an arbitrary $9\times11$ rectangle. How many red squares does it contain? (The sides of all considered rectangles go along the grid lines.)

For a given positive integer $n$, you may perform a series of steps. At each step, you may apply an operation: you may increase your number by one, or if your number is divisible by 2, you may divide your number by 2. Let $\ell(n)$ be the minimum number of operations needed to transform the number $n$ to 1 (for example, $\ell(1) = 0$ and $\ell(7) = 4$). How many positive integers $n$ are there such that $\ell(n) \leq 12$?

[u]Round 1[/u] [b]p1.[/b] Three two-digit numbers are written on a board. One starts with $5$, another with $6$, and the last one with $7$. Annie added the first and the second numbers; Benny added the second and the third numbers; Denny added the third and the first numbers. Could it be that one of these sums is equal to $148$, and the two other sums are three-digit numbers that both start with $12$? [b]p2.[/b] Three rocks, three seashells, and one pearl are placed in identical boxes on a circular plate in the order shown. The lids of the boxes are then closed, and the plate is secretly rotated. You can open one box at a time. What is the smallest number of boxes you need to open to know where the pearl is, no matter how the plate was rotated? [img]https://cdn.artofproblemsolving.com/attachments/0/2/6bb3a2a27f417a84ab9a64100b90b8768f7978.png[/img] [b]p3.[/b] Two detectives, Holmes and Watson, are hunting the thief Raffles in a library, which has the floorplan exactly as shown in the diagram. Holmes and Watson start from the center room marked $D$. Show that no matter where Raffles is or how he moves, Holmes and Watson can find him. Holmes and Watson do not need to stay together. A detective sees Raffles only if they are in the same room. A detective cannot stand in a doorway to see two rooms at the same time. [img]https://cdn.artofproblemsolving.com/attachments/c/1/6812f615e60a36aea922f145a1ffc470d0f1bc.png[/img] [b]p4.[/b] A museum has a $4\times 4$ grid of rooms. Every two rooms that share a wall are connected by a door. Each room contains some paintings. The total number of paintings along any path of $7$ rooms from the lower left to the upper right room is always the same. Furthermore, the total number of paintings along any path of $7$ rooms from the lower right to the upper left room is always the same. The guide states that the museum has exactly $500$ paintings. Show that the guide is mistaken. [img]https://cdn.artofproblemsolving.com/attachments/4/6/bf0185e142cd3f653d4a9c0882d818c55c64e4.png[/img] [b]p5.[/b] The numbers $1–14$ are placed around a circle in some order. You can swap two neighbors if they differ by more than $1$. Is it always possible to rearrange the numbers using swaps so they are ordered clockwise from $1$ to $14$? [u]Round 2[/u] [b]p6.[/b] A triangulation of a regular polygon is a way of drawing line segments between its vertices so that no two segments cross, and the interior of the polygon is divided into triangles. A flip move erases a line segment between two triangles, creating a quadrilateral, and replaces it with the opposite diagonal through that quadrilateral. This results in a new triangulation. [img]https://cdn.artofproblemsolving.com/attachments/a/a/657a7cf2382bab4d03046075c6e128374c72d4.png[/img] Given any two triangulations of a polygon, is it always possible to find a sequence of flip moves that transforms the first one into the second one? [img]https://cdn.artofproblemsolving.com/attachments/0/9/d09a3be9a01610ffc85010d2ac2f5b93fab46a.png[/img] [b]p7.[/b] Is it possible to place the numbers from $1$ to $121$ in an $11\times 11$ table so that numbers that differ by $1$ are in horizontally or vertically adjacent cells and all the perfect squares $(1, 4, 9,..., 121)$ are in one column? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose July of year $ N$ has five Mondays. Which of the following must occur five times in August of year $ N$? (Note: Both months have $ 31$ days.) $ \textbf{(A)}\ \text{Monday} \qquad \textbf{(B)}\ \text{Tuesday} \qquad \textbf{(C)}\ \text{Wednesday} \qquad \textbf{(D)}\ \text{Thursday} \qquad \textbf{(E)}\ \text{Friday}$

On a certain planet, the alien inhabitants are born without any arms, legs, or noses. Every year, on their birthday, each alien randomly grows either an arm, a leg, or a nose, with equal probability for each. After its sixth birthday, the probability that an alien will have at least $2$ arms, at least $2$ legs, and at least $1$ nose on the day is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

The incircle of $\triangle ABC$ is centered at $I$ and is tangent to $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. A circle with radius $2$ is centered at each of $D$, $E$, and $F$. Circle $D$ intersects circle $I$ at points $D_1$ and $D_2$. The points $E_1$, $E_2$, $F_1$, and $F_2$ are defined similarly. If the inradius of $\triangle ABC$ is $5$, what is the ratio of the area of the triangle whose sides are formed by extending $D_1D_2$, $E_1E_2$, and $F_1F_2$ to the area of $\triangle ABC$?

Mike has 12 books, Sean has 9 books, and little Sherry has only 4 books. Find the percentage of these books that Sean has.

An acute triangle $ABC$ in which $AB<AC$ is given. The bisector of $\angle BAC$ crosses $BC$ at $D$. Point $M$ is the midpoint of $BC$. Prove that the line though centers of circles escribed on triangles $ABC$ and $ADM$ is parallel to $AD$.

For what digit $A$ is the numeral $1AA$ a perfect square in base-$5$ and a perfect cube in base-$6$? $\text{(A) }0\qquad\text{(B) }1\qquad\text{(C) }2\qquad\text{(D) }3\qquad\text{(E) }4$

Let $ABC$ be a triangle and let $P$ be a point in its interior. Denote by $D$, $E$, $F$ the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$, respectively. Suppose that \[AP^2 + PD^2 = BP^2 + PE^2 = CP^2 + PF^2.\] Denote by $I_A$, $I_B$, $I_C$ the excenters of the triangle $ABC$. Prove that $P$ is the circumcenter of the triangle $I_AI_BI_C$. [i]Proposed by C.R. Pranesachar, India [/i]

Each of $11$ weights is weighing an integer number of grams. No two weights are equal. It is known that if all these weights or any group of them are placed on a balance then the side with a larger number of weights is always heavier. Prove that at least one weight is heavier than $35$ grams.

Find all pairs of positive integers $(n, k)$ satisfying the equation $$n!+n=n^k.$$

Determine the largest constant $C$ such that $$(x_1 + x_2 + \cdots + x_6)^2 \ge C \cdot (x_1(x_2 + x_3) + x_2(x_3 + x_4) + \cdots + x_6(x_1 + x_2))$$ holds for all real numbers $x_1, x_2, \cdots , x_6$. For this $C$, determine all $x_1, x_2, \cdots x_6$ such that equality holds. (Walther Janous)

Radii of five concentric circles $\omega_0,\omega_1,\omega_2,\omega_3,\omega_4$ form a geometric progression with common ratio $q$ in this order. What is the maximal value of $q$ for which it's possible to draw a broken line $A_0A_1A_2A_3A_4$ consisting of four equal segments such that $A_i$ lies on $\omega_i$ for every $i=\overline{0,4}$? [hide=thanks ]Thanks to the user Vlados021 for translating the problem.[/hide]

Find all pairs of natural numbers $(a,n)$, $a\geq n \geq 2,$ for which $a^n+a-2$ is a power of $2$.

For all real numbers $x$, we denote by $\lfloor x \rfloor$ the largest integer that does not exceed $x$. Find all functions $f$ that are defined on the set of all real numbers, take real values, and satisfy the equality \[f(x + y) = (-1)^{\lfloor y \rfloor} f(x) + (-1)^{\lfloor x \rfloor} f(y)\] for all real numbers $x$ and $y$. [i]Navneel Singhal, India[/i]

From given positive numbers, the following infinite sequence is defined: $a_1$ is the sum of all original numbers, $a_2$ is the sum of the squares of all original numbers, $a_3$ is the sum of the cubes of all original numbers, and so on ($a_k$ is the sum of the $k$-th powers of all original numbers). a) Can it happen that $a_1 > a_2 > a_3 > a_4 > a_5$ and $a_5 < a_6 < a_7 < \ldots$? (4 points) b) Can it happen that $a_1 < a_2 < a_3 < a_4 < a_5$ and $a_5 > a_6 > a_7 > \ldots$? (4 points) [i](Alexey Tolpygo)[/i]

Let $a_{1}={11}^{11}$, $a_{2}={12}^{12}$, $a_{3}={13}^{13}$, and \[a_{n}= \vert a_{n-1}-a_{n-2}\vert+\vert a_{n-2}-a_{n-3}\vert, n \ge 4.\] Determine $a_{{14}^{14}}$.

Find all pairs of positive integers $a, b$ such that $\sqrt{a+2\sqrt{b}}=\sqrt{a-2\sqrt{b}}+\sqrt{b}$ .

Find all real solutions to the equation \[| | | | | | |x^2 -x - 1| - 3| - 5| - 7| - 9| - 11| - 13| = x^2 - 2x - 48.\]

Misha and Sahsa play a game on a $100\times 100$ chessboard. First, Sasha places $50$ kings on the board, and Misha places a rook, and then they move in turns, as following (Sasha begins): At his move, Sasha moves each of the kings one square in any direction, and Misha can move the rook on the horizontal or vertical any number of squares. The kings cannot be captured or stepped over. Sasha's purpose is to capture the rook, and Misha's is to avoid capture. Is there a winning strategy available for Sasha?

The number $2020$ has three different prime factors. What is their sum?

[u]Round 1[/u] [b]p1.[/b] Is it possible to draw some number of diagonals in a convex hexagon so that every diagonal crosses EXACTLY three others in the interior of the hexagon? (Diagonals that touch at one of the corners of the hexagon DO NOT count as crossing.) [b]p2.[/b] A $ 3\times 3$ square grid is filled with positive numbers so that (a) the product of the numbers in every row is $1$, (b) the product of the numbers in every column is $1$, (c) the product of the numbers in any of the four $2\times 2$ squares is $2$. What is the middle number in the grid? Find all possible answers and show that there are no others. [b]p3.[/b] Each letter in $HAGRID$'s name represents a distinct digit between $0$ and $9$. Show that $$HAGRID \times H \times A\times G\times R\times I\times D$$ is divisible by $3$. (For example, if $H=1$, $A=2$, $G=3$, $R = 4$, $I = 5$, $D = 64$, then $HAGRID \times H \times A\times G\times R\times I\times D= 123456\times 1\times2\times3\times4\times5\times 6$). [b]p4.[/b] You walk into a room and find five boxes sitting on a table. Each box contains some number of coins, and you can see how many coins are in each box. In the corner of the room, there is a large pile of coins. You can take two coins at a time from the pile and place them in different boxes. If you can add coins to boxes in this way as many times as you like, can you guarantee that each box on the table will eventually contain the same number of coins? [b]p5.[/b] Alex, Bob and Chad are playing a table tennis tournament. During each game, two boys are playing each other and one is resting. In the next game the boy who lost a game goes to rest, and the boy who was resting plays the winner. By the end of tournament, Alex played a total of $10$ games, Bob played $15$ games, and Chad played $17$ games. Who lost the second game? [u]Round 2[/u] [b]p6.[/b] After going for a swim in his vault of gold coins, Scrooge McDuck decides he wants to try to arrange some of his gold coins on a table so that every coin he places on the table touches exactly three others. Can he possibly do this? You need to justify your answer. (Assume the gold coins are circular, and that they all have the same size. Coins must be laid at on the table, and no two of them can overlap.) [b]p7.[/b] You have a deck of $50$ cards, each of which is labeled with a number between $1$ and $25$. In the deck, there are exactly two cards with each label. The cards are shuffled and dealt to $25$ students who are sitting at a round table, and each student receives two cards. The students will now play a game. On every move of the game, each student takes the card with the smaller number out of his or her hand and passes it to the person on his/her right. Each student makes this move at the same time so that everyone always has exactly two cards. The game continues until some student has a pair of cards with the same number. Show that this game will eventually end. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].